Document Open Access Logo

Dynamics for the Mean-field Random-cluster Model

Authors Antonio Blanca, Alistair Sinclair

PDF
Thumbnail PDF

File

LIPIcs.APPROX-RANDOM.2015.528.pdf
  • Filesize: 0.57 MB
  • 16 pages

Document Identifiers

Author Details

Antonio Blanca
Alistair Sinclair

Cite As Get BibTex

Antonio Blanca and Alistair Sinclair. Dynamics for the Mean-field Random-cluster Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 528-543, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.528

Abstract

The random-cluster model has been widely studied as a unifying framework for random graphs, spin systems and random spanning trees, but its dynamics have so far largely resisted analysis. In this paper we study a natural non-local Markov chain known as the Chayes-Machta dynamics for the mean-field case of the random-cluster model, and identify a critical regime (lambda_s,lambda_S) of the model parameter lambda in which the dynamics undergoes an exponential slowdown. Namely, we prove that the mixing time is Theta(log n) if lambda is not in [lambda_s,lambda_S], and e^Omega(sqrt{n}) when lambda is in (lambda_s,lambda_S). These results hold for all values of the second model parameter q > 1. In addition, we prove that the local heat-bath dynamics undergoes a similar exponential slowdown in (lambda_s,lambda_S).

Subject Classification

Keywords
  • random-cluster model
  • random graphs
  • Markov chains
  • statistical physics
  • dynamics

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. V. Beffara and H. Duminil-Copin. The self-dual point of the two-dimensional random-cluster model is critical for q ≥ 1. Probability Theory and Related Fields, 153:511-542, 2012. Google Scholar
  2. A. Blanca and A. Sinclair. Dynamics for the mean-field random-cluster model, 2014. Preprint. arXiv:1412.6180v1 [math.PR]. Google Scholar
  3. B. Bollobás, G. Grimmett, and S. Janson. The random-cluster model on the complete graph. Probability Theory and Related Fields, 104(3):283-317, 1996. Google Scholar
  4. C. Borgs, J. Chayes, and P. Tetali. Swendsen-Wang algorithm at the Potts transition point. Probability Theory and Related Fields, 152:509-557, 2012. Google Scholar
  5. C. Borgs, A. Frieze, J.H. Kim, P. Tetali, E. Vigoda, and V. Vu. Torpid mixing of some Monte Carlo Markov chain algorithms in statistical physics. Proceedings of the 40th Annual Symposium on Foundations of Computer Science (FOCS), pages 218-229, 1999. Google Scholar
  6. L. Chayes and J. Machta. Graphical representations and cluster algorithms II. Physica A, 254:477-516, 1998. Google Scholar
  7. C. Cooper, M.E. Dyer, A.M. Frieze, and R. Rue. Mixing properties of the Swendsen-Wang process on the complete graph and narrow grids. Journal of Mathematical Physics, 41:1499-1527, 2000. Google Scholar
  8. P. Cuff, J. Ding, O. Louidor, E. Lubetzky, Y. Peres, and A. Sly. Glauber dynamics for the mean-field Potts model. Journal of Statistical Physics, 149(3):432-477, 2012. Google Scholar
  9. R.G. Edwards and A.D. Sokal. Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. Physical Review D, 38(6):2009-2012, 1988. Google Scholar
  10. C.M. Fortuin and P.W. Kasteleyn. On the random-cluster model I. Introduction and relation to other models. Physica, 57(4):536-564, 1972. Google Scholar
  11. A. Galanis, D. Štefankovič, and E. Vigoda. Swendsen-Wang Algorithm on the Mean-Field Potts Model, 2015. Preprint. arXiv:1502.06593v1 [cs.DM]. Google Scholar
  12. Q. Ge and D. Štefankovič. A graph polynomial for independent sets of bipartite graphs. Combinatorics, Probability and Computing, 21(5):695-714, 2012. Google Scholar
  13. V.K. Gore and M.R. Jerrum. The Swendsen-Wang process does not always mix rapidly. Journal of Statistical Physics, 97(1-2):67-86, 1999. Google Scholar
  14. G.R. Grimmett. The Random-Cluster Model, volume 333 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2006. Google Scholar
  15. M. Huber. A bounding chain for Swendsen-Wang. Random Structures & Algorithms, 22(1):43-59, 2003. Google Scholar
  16. G. Kirchhoff. Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Strome gefuhrt wird. Annalen der Physik und Chemie, pages 497-508, 1847. Google Scholar
  17. D.A. Levin, Y. Peres, and E.L. Wilmer. Markov Chains and Mixing Times. American Mathematical Society, 2008. Google Scholar
  18. Y. Long, A. Nachmias, W. Ning, and Y. Peres. A power law of order 1/4 for critical mean-field Swendsen-Wang dynamics. Memoirs of the American Mathematical Society, 232(1092), 2011. Google Scholar
  19. M.J. Luczak and T. Łuczak. The phase transition in the cluster-scaled model of a random graph. Random Structures & Algorithms, 28(2):215-246, 2006. Google Scholar
  20. R.H. Swendsen and J.S. Wang. Nonuniversal critical dynamics in Monte Carlo simulations. Physical Review Letters, 58:86-88, 1987. Google Scholar
  21. M. Ullrich. Rapid mixing of Swendsen-Wang dynamics in two dimensions. Dissertationes Mathematicae, 502, 2014. Google Scholar
  22. M. Ullrich. Swendsen-Wang is faster than single-bond dynamics. SIAM Journal on Discrete Mathematics, 28(1):37-48, 2014. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact